Properties

Degree 1
Conductor 293
Sign $-0.891 - 0.452i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.798 − 0.601i)2-s + (−0.683 − 0.729i)3-s + (0.276 + 0.961i)4-s + (−0.150 + 0.988i)5-s + (0.107 + 0.994i)6-s + (−0.744 + 0.668i)7-s + (0.357 − 0.933i)8-s + (−0.0645 + 0.997i)9-s + (0.714 − 0.699i)10-s + (−0.397 + 0.917i)11-s + (0.512 − 0.858i)12-s + (−0.976 + 0.213i)13-s + (0.996 − 0.0859i)14-s + (0.823 − 0.566i)15-s + (−0.847 + 0.530i)16-s + (−0.798 − 0.601i)17-s + ⋯
L(s,χ)  = 1  + (−0.798 − 0.601i)2-s + (−0.683 − 0.729i)3-s + (0.276 + 0.961i)4-s + (−0.150 + 0.988i)5-s + (0.107 + 0.994i)6-s + (−0.744 + 0.668i)7-s + (0.357 − 0.933i)8-s + (−0.0645 + 0.997i)9-s + (0.714 − 0.699i)10-s + (−0.397 + 0.917i)11-s + (0.512 − 0.858i)12-s + (−0.976 + 0.213i)13-s + (0.996 − 0.0859i)14-s + (0.823 − 0.566i)15-s + (−0.847 + 0.530i)16-s + (−0.798 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.891 - 0.452i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.891 - 0.452i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $-0.891 - 0.452i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (40, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ -0.891 - 0.452i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.03601104790 - 0.1507011351i$
$L(\frac12,\chi)$  $\approx$  $0.03601104790 - 0.1507011351i$
$L(\chi,1)$  $\approx$  0.3969029835 - 0.09661642126i
$L(1,\chi)$  $\approx$  0.3969029835 - 0.09661642126i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−26.04758634371755185125216613910, −24.949020157024509230505955710662, −23.92573266197059108692165455965, −23.4998599251690845288306501519, −22.330408723872800486299979415964, −21.32846591742902134156515650251, −20.06477093953590987111993440867, −19.725793187006083107255515686, −18.332041294687230828323786147529, −17.250853029066366282440609389104, −16.71413332589381950757976838093, −16.029679736636741101075772879755, −15.33624461197506788197767326073, −13.9864163726237931929939753421, −12.80678793520058580552069457681, −11.60528152794667729857530186059, −10.57682990591672274923404295675, −9.76962637510826417420792560765, −8.9718518579348554807521371812, −7.84471504629138582188611283982, −6.652635720323283711292057133629, −5.5754677719135608065157776552, −4.81376796800239455441035961703, −3.40586193919832512750640577035, −1.16794220520865109818238027127, 0.15950470755051451943742691249, 2.29938659689415290250579467226, 2.65330871239211408018916407873, 4.50818461931110105766813700250, 6.12791413289488137156141477425, 7.124411833261797411256293291605, 7.617741832479012362094879795637, 9.22971961296209826681803535473, 10.07812036491916789521132416571, 11.08470508997690034727210275348, 11.92386748495212737394862594484, 12.61556078865824825517288645681, 13.64358907125705709573680384797, 15.19894171073380021957597086069, 16.07891687163094612515896121675, 17.2280808819908944486634949815, 18.04555742919685334332932077574, 18.608777565646446200425324112726, 19.38679277035024789577578012682, 20.20263574806432380021742338674, 21.69677141950258328627531345763, 22.43028358144513222324571477318, 22.84831554769053697766336065128, 24.41712048953136180338969326686, 25.14233225488919339834408401404

Graph of the $Z$-function along the critical line